The members of \(\Psi \)  are nonzero scalar multiples \(\alpha _{\mathfrak {e}}\psi _{\mathfrak {e}}\)  of the rank-1 forms \(\psi _{\mathfrak {e}}\) , one for
each edge. Let \(\Sigma \psi =\sum _{\mathfrak {\psi \in \Psi }}\psi \) .
The nonzero entries in column \(\mathfrak {e}\)  of \(\left[ \Sigma \psi \right] \) 
are \(\alpha _{\mathfrak {e}}\)  in the rows indexed by the vertices of \(\mathfrak {e}\) . The incidence matrix \(I(G)\)  is the matrix indexed the
same way as \(\left[ \Sigma \psi \right] \) , but with 1 in row \(x\)  and
column \(\mathfrak {e}\)  if \(x\)  is a vertex of \(\mathfrak {e}\) , and 0 if not
[1]}. Thus \(\left[ \Sigma \psi \right] \)  is obtained from \(I(G)\) 
by scaling the columns, column \(\mathfrak {e}\)  by \(\alpha _{\mathfrak {e}}\) .
In particular, \(\mathrm {rank}\left[ \Sigma \psi \right] =\mathrm {rank}I(G)\) .
